Theil index estimation by means of the influence function with an application to income surveys

By assuming the design-based paradigm, an analysis of the Theil index and its estimation is carried out. First, by expressing the population Theil index as a statistical functional, we obtain its influence function and prove the corresponding properties. We also provide some new results on the influence function of the Gini index which are suitable for a methodological comparison of the two inequality measures. Subsequently, on the basis of these findings, we introduce estimators of the Theil index and its variance. A confidence band for the Theil index influence function is also proposed. Using a simulation study, we show that the variance estimator has suitable performance in terms of bias and provides confidence intervals with adequate coverage. The suggested variance estimation outperforms the corresponding methods based on nonparametric and parametric bootstrap. An application of our achievements is considered using the data from the ‘Survey on Vulnerability to Poverty’ conducted in 2021 in Tuscany (Italy). This survey is designed to obtain reliable estimates at a high disaggregated level and allow the estimation of the Theil index by distinguishing between production areas and provinces. The results highlight an increasing inequality moving from touristic areas to those where production is based on industries.